In Weinberg's book of Qft, chapter 2 of volume 1, he uses the eigenstates of the four-momentum to construct the unitary irreducible representations of the Poincare group. My question is, since $P^\mu,P^\mu P_\mu$ and $W^\mu W_\mu$ forms a complete set of commuting operators, why didn't he use the common eigenstates of these 3 operators to label states?

Since I know the state $\Psi_{p,\sigma}$ is an eigenstate of $P^\mu$, it must also be an eigenstate of $P^\mu P_\mu$ with eigenvalue $m^2$. But what about the action of $W^2$ on states $\Psi_{p,\sigma}$? Does he implicitly assume that $\sigma$ labels the eigenvalue of $W^2$?


1 Answer 1


The way irreducible representations of the proper Poincaré group are studied is through the machinery of induced representations. Since the Poincaré group (or more precisely its 2-cover) has a semidirect product structure by an Abelian group $N$, one can used a special case of the general theory of induced representations, known as Wigner's little group method. In order for this to work one has to determine a Borel section of the Pontryagin dual of the Abelian group ($(\mathbb R,+)$ in this case, whose dual represents momentum space) and this consists, e.g. of the whole time axis, the positive $x$-axis, and the points $(1,1,0,0)$ and $(1,-1,0,0)$.

On this Borel section you can fix some reference points. Equivalence classes are then distinguished by orbits of this reference points under the proper Lorentz group, i.e. the other factor of the semidirect product. The little groups are just the stabilisers of these points. Since the Poincaré group is of type I, when you determine an irreducible unitary representation of the little group $H_\chi$ and you fix a character for the Abelian group $N$, the theory of induced representations gives you a unitary representation of the whole Poicaré group out of a unitary representation of $H_\chi\ltimes N$.

Of course, once you have an irreducible representation, every central operator reduces to a multiple of the identity, so that the different particles are labelled by the values of $m$ and $\sigma$, i.e. square root of the eigenvalues of the Casimir operators $p_\mu p^\mu$ and $W_\mu W^\mu$, in the kinematical sense of Wigner.


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